The oil well drilling industry lacks a system for drilling an elongated, small diameter, lateral branch borehole at a selected depth from a main well, where the lateral branch borehole extends a significant distance from the main well, and the transition from main well to branch borehole has a very small radius of curvature. For a typical main well having an internal diameter of approximately 15 cm, a suitable system would drill a branch borehole having a curved transition portion with radius of curvature less than 5 meters, and would then drill a lateral branch borehole, having a diameter of approximately 60 mm, for a distance of approximately 100 meters. Currently available systems for drilling lateral branch boreholes at a selected depth from a main well do not provide this combination of capabilities. A system having this combination of capabilities would be very valuable.